Question

Each of $20$ balls is tossed independently and at random into one of $5$ bins. Let $p$ be the probability that some bin ends up with $3$ balls, another with $5$ balls, and the other three with $4$ balls each. Let $q$ be the probability that every bin ends up with $4$ balls. What is $\dfrac{p}{q}?$

$1$

$4$

$8$

$12$

$16$

Solution:

Both probabilities divide by $5^{20},$ so $\dfrac{p}{q}$ is a ratio of arrangement counts.

For $q,$ all bins have $4:$ $\dfrac{20!}{(4!)^5}.$ For $p,$ choose which bin has $3$ and which has $5$ in $5\cdot 4 = 20$ ways, times $\dfrac{20!}{3!\,5!\,(4!)^3}.$ Therefore $\frac{p}{q} = 20 \cdot \frac{(4!)^5}{3!\,5!\,(4!)^3} = 20 \cdot \frac{(4!)^2}{3!\,5!} = 20 \cdot \frac{576}{720} = 16.$

Thus, the correct answer is E.

2021 AMC 12A Fall Problem 18

Solution:

Problem 18 in Other Years